For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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3-Space Vertices of a Parallelogram

The points (1, -2, 4), (3, 5, 7) and (4, 6, 8) are three of four vertices of parallelogram ABCD. Explain why there are three possibilities for the location of the fourth vertex, and find the three ...
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3answers
48 views

Two Points with Infinite Distance

I have tried looking for an answer to this, but can't seem to find anything. Is it possible on a standard Cartesian grid to define two (or more) points with infinite or undefined distance from each ...
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0answers
71 views

What is the geometrical interpretation of Cartier Divisors?

Definition: Let $(s, \mathcal{L})$ be a pair where $s$ is a rational section of the line bundle $\mathcal{L}$. The Cartier divisor is defined as this pair $(s, \mathcal{L})$. My question: What is ...
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6answers
100 views

Three positive numbers a, b, c satisfy $a^2 + b^2 = c^2$; is it necessarily true that there exists a right triangle with side lengths a,b and c?

If so, how could you go about constructing it? If not, why not? I am new to proofs and I was reading a book where they posed this question. I understand that if we are given any right triangle, the ...
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2answers
15 views

Finding an unknown coordinate

ADE is a straight line . AE : AD = 3 : 2 Find coordinates of E My workings Let E ( X , Y) Gradient AD = Gradient of AE $ 1/3 = X - 3 / Y - 1 $ From here I'm not too sure on how to carry on.. ...
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1answer
16 views

Minimum number of objects to cover a given area, given a total perimeter

So I am given a circular area 10 square units, and I am given a length, 6 units which the total perimeters of all the shapes must add up to. All shapes are counted separately, so if I have 2 squares ...
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4answers
73 views

Problem based on proving equal triangles in area

In the figure, $ABCD$ is a parallelogram. If $O$ be any point on $BD$ then prove that $$\triangle OAB=\triangle OAD+\triangle OAC$$ My Attempt $$\triangle ADB=\triangle BDC$$ $$\triangle OAM=\...
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2answers
59 views

Counting faces of each type in a rhombicosidodecahedron

If I know that, in a rhombicosidodecahedron, at every vertex one triangle, one pentagon, and two squares meet, then how can I compute the number of faces and edges that are needed to build it? There ...
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1answer
29 views

How to rotate a 3D vector on the surface of a plane by a known angle?

Available data The plane β which is defined by a normal vector n and point P. The vector v which lies on the surface of the plane.(the angle between v and n is 90 degrees). The angle α to which v ...
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3answers
58 views

Distance from a set to a point

There is this exercise I cannot understand well. It asks me for the distance between this set in $\mathbb{R}^3$ $$U = \{(x, y, z)\ |\ ax + y - 2z = 0, z = 0 \}$$ and the point $(0, b, 1)$. Also it ...
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2answers
133 views

If the area of $ ABP$ is $ 192 $ find $ PA*PC $

Let $ABCD$ be an isosceles trapezium with bases $ AB=32 $ and $CD=18$. Inside $ABCD$ there's a point $P$ such that $ \angle PAD= \angle PBA $ and $ \angle PDA =\angle PCD $. If the area of $ ABP$ is $ ...
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1answer
33 views

Find radius of circle (or sphere) given segment area (or cap volume) and chord length

The goal is to design a container (partial sphere) of given volume which attached to a plane via a port of a given radius. I believe this can be done as follows but the calculation is causing me ...
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1answer
41 views

Quadrics and quadratic forms.

I'm learning about quadrics. My textbook says that a quadric is a quadratic form $X^TAX = 0$ with $A$ en symmetric matrix and $X$ the homogeneous coordinates in a orthonormal basis. But if we're ...
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1answer
38 views

What norm Induced inner product?

Is inner product come from a norm? Or norm comes from inner product? How to prove this is true? Just want to know what comes from what, the chicken or the egg came first kind of thing.
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1answer
80 views

Upper bound on the minimum distance between $N$ points chosen inside the unit circle?

I guess this is a well-known problem but I'm not sure where to find it on the web. $N \ge 2$ points are chosen in the interior or the boundary of the unit circle. What is the best upper bound on the ...
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1answer
59 views

Proof of an inequality involving general triangles

In any triangle the following inequality holds: $$\dfrac{9abc}{a+b+c}\ge4S\sqrt{3}$$ where $a,b,c$ are the sides of the triangle and $S$ the area. How the previous inequality can be proven?
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168 views

If every five point subset of a metric space can be isometrically embedded in the plane then is it possible for the metric space also?

Let $X$ be a metric space with at least $5$ points such that any five point subset of $X$ can be isometrically embedded in $\mathbb R^2$ , then is it true that $X$ can also be isometrically embedded ...
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1answer
14 views

Planes through $OX$ and $OY$ include an angle $\alpha,$ show that their line of intersection lies on the cone $z^2(x^2+y^2+z^2)=x^2y^2\tan^2\alpha$

Planes through $OX$ and $OY$ include an angle $\alpha,$ show that their line of intersection lies on the cone $z^2(x^2+y^2+z^2)=x^2y^2\tan^2\alpha$ The lines of intersection of the planes through $...
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1answer
75 views

Show that the vertex lies on the surface $z^2(\frac{x}{a}+\frac{y}{b})=4(x^2+y^2)$

Two cones with a common vertex pass through the curves $z^2=4ax,y=0$ and $z^2=4by,x=0.$ The plane $z=0$ meets them in two conics which intersect in four concyclic points.Show that the vertex lies on ...
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0answers
29 views

Coxeter's “Introduction to Geometry” recommendation

What mathematical background does one need for Coxeter's "Introduction to Geometry"? Is the text suitable for self-study?
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0answers
32 views

Theorems of euclidean geometry as invariable properties of geometric configurations

Is there some book, or systematic theory, that proves theorems of euclidean geometry by viewing them as invariable properties of certain geometric configurations ? So that from an easy special case, ...
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2answers
63 views

Prove that the equation of the cone $yz(\frac{b}{c})+zx(\frac{c}{a}+\frac{a}{c})+xy(\frac{a}{b}+\frac{b}{a})=0$

The plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ cuts the coordinate axes in $A,B,C.$Prove that the lines passing through the origin and intersecting the circle $ABC$ generate the cone $yz(\frac{b}{c}...
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1answer
34 views

Calculate Section (area) of N-Dimensional Tube

I have the following n-dimensional shape $1=\sum_{i=1}^{n}a_{i}x_{i}^{2}$ where $a_{i}>0$ and I'd like to calculate the cross-section area inside. Any suggestion? Note: I call it an $n$-...
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1answer
96 views

Maximizing area of a pentagon

Suppose $a,b,c,d,e$ are pairwise distinct positive integers. Consider a pentagon with sides $a,b,c,d,e$ and with angles maximizing its area (we assume that a pentagon with such sides exists). It is ...
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1answer
53 views

Longest distance to the foci or the center that a point within the ellipse can be?

Given an ellipse $E$ (with the foci $f_1$ and $f_2$ and the center $c$), and a point $p$, which is the maximum distance that $p$ can be to all these 3 points to be within the ellipse $E$? I.e., which ...
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0answers
69 views

Interpreting the volume of parallelotope in $\mathbb{C}^n$

I am confused about the interpretation of the volume of parallelotope generated by the set of vectors $<a_1, \ldots, a_n>$ in $\mathbb{C}^n$. Squared Gram determinant gives the real value , ...
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0answers
23 views

Is there an expanding rectangular space filling curve 'growing' around a point in space?

Do there exist 'expanding' 3D rectangular space filling curves? (The actual problem: I'm writing a block-based voxel engine where chunks should be loaded depending on their distance from the player. ...
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1answer
32 views

Distance between a point in $3$-D and the line $x=0$ ($y$-axis) [closed]

Find the the perpendicular distance (which I assume will be the shortest distance both in $3$-D and $2$-D (if not then please find both perpendicular distance and shortest distance) of the $y$-axis ($...
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0answers
8 views

Show that the $ZX-$ plane cuts it in the curve $F(\frac{bx}{x-a},\frac{cx-az}{x-a})=0,y=0.$

The vertex of the cone is $(a,b,c)$ and $YZ$-plane cuts it in the curve $F(y,z)=0,x=0$.Show that the $ZX-$ plane cuts it in the curve $F(\frac{bx}{x-a},\frac{cx-az}{x-a})=0,y=0.$ Let the equation ...
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0answers
29 views

Counting balls in face centred cubic close packing

Possibly too easy for stack exchange, but... Consider a cubic close packing, or face centred cubic, arrangement of balls or radius $1$ in dimension $3$. Suppose that the origin is the centre of one ...
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0answers
17 views

The section of a cone whose vertex is $P$ and guiding curve $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,z=0$ by the plane $x=0$ is rectangular hyperbola.

The section of a cone whose vertex is $P$ and guiding curve the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,z=0$ by the plane $x=0$ is rectangular hyperbola.Show that the locus of $P$ is $\frac{x^2}{a^...
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1answer
49 views

Why is it against common sense?

The question is this. A man can walk at speeds of 6kmph uphill, 7.5kmph along level surface and 10kmph downhill. He travels from A to B in 3 hours and from B to A in 1 hour. What is distance AB? I ...
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1answer
30 views

Euler's Formula: $V-E+F=2$ by using spheric triangles

I just have a question to a proof found here: https://nrich.maths.org/1384 At one point it says: As eight copies of $\triangle$ will fill the sphere without overlapping. Why this? Why can I "...
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3answers
67 views

Area of the triangle formed by circumcenter, incenter and orthocenter

Lets say we have $\triangle$$ABC$ having $O,I,H$ as its circumcenter, incenter and orthocenter. How can I go on finding the area of the $\triangle$$HOI$. I thought of doing the question using the ...
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1answer
33 views

Prove the inequality of area of convex polygon X is A is less than or equal to $\frac{\pi d^2}{ 4}$ [closed]

I want to prove that convex polygon X in the plane has diameter d, its area is less than or equal to $\frac{\pi d^2}{ 4}$.
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4answers
63 views

Show that the unit sphere is connected [duplicate]

I need to show that $\{(x,y,z)\in\mathbb{R}^{3}:x^2+y^2+z^2 = 1\}$ is connected. Intuitively I understand that it is path connected and, therefore, connected. However, I don't understand how I would ...
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1answer
64 views

Splitting Line Segments and Finding Expected Value

Consider a line segment which has a length of $2n-3$. It is split into $n$ segments at random. It is guaranteed that $n\ge 3$ and $n\in \mathbb{Z}$. These smaller lines are then used as the sides of a ...
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0answers
23 views

I have a convex hull (generated from a library) in 3D. I only have the vertices. How do I compute the volume of the hull.

I have a library (quickhull in C++) that I am using to create a hull from a set of points. I am able to see the vertices of the hull but not the facets. I would like to compute the volume of the hull. ...
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1answer
52 views

Circle packing – How to get the minimum length?

In an a past admission paper from a local university, I came across a problem I couldn't solve. Given $n$ circles with their respective radii $r_1, r_2, \dotsc , r_n,$ we are to find the minimum ...
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What is the minimum number of sets of Euler angles to cover $SO(3)$?

This is a question I was asked to answer from a drone-robotics check assignment. What is the minimum number of sets of Euler angles to cover $SO(3)$? $$SO(3)=\{R\in\mathbb{R}^{3\times 3}|R^TR=RR^T=I\...
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0answers
48 views

What curves will satisfy this very intersting property?

Let $c_1,c_2\subset\mathbb R^2$ be differentiable curves. Given that for any rigid transformation $E$ (i.e. combination of reflections, translations, rotations), if $c_1,E(c_2)$ intersect ...
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0answers
76 views

cutting an equilateral triangle to $n$ equal pieces

We have an equilateral triangle and we want to cut it into $n$ equal pieces. For which $n$ is it possible? My Attempt: I found these possible numbers $2,3,4,6$ and also I proved every $n$ of the ...
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2answers
58 views

How many sphere on the boundary of a big sphere?

I don't know exactly how to ask this in a comprehensible way. I am trying to find a solution to my problem which is to find how many sphere of radius r are lying on the boundary (which means that in ...
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0answers
29 views

Properties of polyhedron solving constrained max problem

This is a question for people who don't have trouble to think in more than two dimensions. Don't hesitate to ask clarifying questions! Let us suppose we have $n$ random variables $X_i$ that are iid ...
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0answers
21 views

Smallest enclosing cylinder

I have a set of 3D points that approximately lie on a cylinder. This cylinder is straight and can be oriented in any direction. I would like to compute the minimal enclosing cylinder for the set; taht ...
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0answers
36 views

Given set of points in 3D, find group of points closest to each other

Given a set of any 8 points in 3D space. I want to find a subset of points that are closest to each other. Application: Assume in a 3D space, I have any 8 colors(represented in RGB). I know how to ...
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0answers
23 views

meromorphic function on torus

Consider the familly of meromorphic function on the square torus (endowed with the corresponding complex structure) with $p$ simple poles and $p$ simple zeros and $L^1$-norm equal to $1$ : $\mathcal ...
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1answer
64 views

In $\triangle ABC$, if $\tan A$, $\tan B$, $\tan C$ are in harmonic progression, then what is the minimum value of $\cot \frac{B}{2}$?

In a $\triangle ABC$, if $\tan A$, $\tan B$, $\tan C$ are in harmonic progression, then what is the minimum value of $\cot(B/2)$? $\bf{My\; Try::}$ Here $A+B+C=\pi\;,$ Then $\tan A+\tan B+\tan C=\...